This wouldn’t be immediate, but you could set up something in Integromat fairly easily. Set a trigger that looks for the change in that {Approval Status} field in the Applicants base, add a new record in the Members base matching the original data, then delete the original record. A three-step scenario.
I say this wouldn’t be immediate because the trigger mechanism in Integromat only checks for changes every 15 minutes on their free account. However, if you add a filter to the Airtable view where you make these changes so that it only shows records where the status is not “Approved,” approving a record would immediately hide it from view, so it likely wouldn’t matter if it gets moved right away or 15 minutes later.
If you want to make it immediate, you can build a webhook triggered by a link created in a formula field. If you want more details on that, let me know. I set that up for myself the other day, and it works like a charm. Click the link, and the scenario runs immediately, so you’d just click that in lieu of relying on the status change.